Journal of Fractional Calculus and Nonlinear Systems

Nonlinear viscoelastic Petrovsky equation with fractional damped: Existence and blow-up

Erhan Pişkin — 2025-12-28

In this article, we study a nonlinear viscoelastic Petrovsky equation with fractional damping. First, we establish the existence of a local weak solution by using semigroup theory. Then, we prove the blow up of the solution under suitable conditions.

Stability and Simulation of Fractional Dynamics in Nonhomogeneous Media via Generalized Quantum-Hadamard Operators

Rabha W. Ibrahim — 2025-12-28

In this study, we extend the concept of the quantum Gamma function (q-Gamma) by introducing a new (q,\tau)-deformed Gamma function. This generalization allows us to construct an enriched family of Hadamard-type fractional operators, which we then apply to the analysis of memory and decoherence in open quantum systems. The inclusion of the deformation parameter q together with the delay-like scaling parameter \tau makes the proposed (q,\tau)-Hadamard framework particularly suited to capture nonlocal and non-Markovian effects, thereby offering a flexible tool for describing structured reservoirs and anomalous dissipation. To investigate the analytical consequences, we employ a (q,\tau)-Mittag-Leffler function, through which explicit solutions are obtained for a class of fractional differential equations, with particular emphasis on population dynamics models. These solutions reveal a variety of memory-driven features, including sub-exponential decay of coherence and the occurrence of revival phenomena. Both the fractional order alpha and the deformation parameters play a decisive role in shaping the temporal behavior. Beyond population models, the framework also provides insights into fractional quantum master equations, quantum walks with fractional memory, and noise effects in quantum information processing.

Generalized Mittag-Leffler Function: Properties in Fractional Calculus and Integral Transforms

Maged G. Bin-Saad — 2025-12-28

In this work, we introduce a new generalized Mittag-Leffler function defined via the extended Beta function and establish its integral and differential representations. Several fundamental properties are derived, including differentiation formulas and the Beta, Laplace, and Mellin transforms, together with relationships to the Wright function and generalized hypergeometric functions. Furthermore, the behavior of the associated Riemann-Liouville fractional integrals and derivatives of the proposed function is investigated. A number of interesting special cases are also presented to illustrate the generality and unifying nature of the main results. In the final section, we discuss potential applications of the newly defined Mittag-Leffler function, demonstrate its use in solving a fractional kinetic equation, and outline possible directions for future research.

Fractional Corrected Simpson's Second Formula Type Inequalities via Extended s-Convexity

Badreddine Meftah — 2025-12-28

In this paper, we establish new fractional variants of the corrected Simpson’s second formula type inequalities by leveraging the concept of extended s-convexity. To achieve this, we first derive a novel integral identity involving Riemann--Liouville fractional integrals, which serves as a fundamental auxiliary result. Building upon this identity, we obtain several inequalities for functions whose first-order derivatives satisfy the extended s-convexity condition on a given interval. Furthermore, we demonstrate the practical relevance of our theoretical findings by applying them to derive estimates for special means. These applications highlight the utility of our inequalities in numerical analysis and approximation theory.

Fractional Powersets and SuperHyperStructures: Toward a Framework for Fractional Set Theory and Discrete Hierarchical Systems

Takaaki Fujita — 2025-12-28

Hyperstructures build on powersets to model multivalued relations on a base set; SuperHyperstructures iterate the powerset to capture layered hierarchies and richer composition. Prior work typically fixes the iteration height to a nonnegative integer. This paper asks whether fractional, inverse, and complex (including imaginary) “heights" can be incorporated coherently. We introduce the notions of an m-root powerset (peeling a specified number of subset layers), a negative powerset (a partial inverse of iterated powersets under a given presentation), and a complex-height powerset defined at the level of observables via operator-theoretic interpolation. We characterize when these operators are well defined—by exponential-tower size conditions in the finite case and by the beth hierarchy in the infinite case—and establish exact inverse laws on their natural domains.
Lifting from carriers to operations, we obtain root and negative SuperHyperStructures that preserve incidence, compose naturally, and recover the original structures after the appropriate number of lifts. Conceptually, the framework provides a principled, continuous interpolation across hierarchical levels and a reversible mechanism for descending them, suggesting applications to discrete modeling, policy design, and multi-resolution analysis.

Analysis of a Fractional Nonlinear SIR Model with Atangana-Baleanu Derivatives

Mohamed Menad — 2025-12-28

We present a fractional nonlinear SIR epidemic model based on the Atangana--Baleanu derivative in the Caputo sense. By incorporating memory and non-local effects, the model offers a more realistic description of disease transmission than classical integer-order formulations. Existence, uniqueness, and Hyers--Ulam stability are established using fixed point theory and generalized Grönwall inequalities, while equilibrium analysis highlights the role of the basic reproduction number. A stable Adams-Bashforth-Moulton predictor--corrector scheme is developed, and numerical experiments confirm accuracy, convergence, and the impact of fractional dynamics on epidemic peaks and persistence. These results underscore the value of fractional operators in epidemiology and point toward integration with artificial intelligence for predictive health modeling.